Another issue is that of “true” vs. “fake” rank; this will be defined in more detail, with examples, in Section REF . Here we briefly say that a pair \(L\) , \(M\)  of commuting operators whose orders are both divisible by \(r\) , is called a “true rank \(r\)  pair” if \(r\)  is the rank of the algebra \( {\bf C}[L,M]\) . We prove in Theorem REF  that BC pairs are true-rank pairs. Of course, not every true-rank pair is a BC pair and, in the process of searching for new true-rank pairs, by means of Grünbaum's approach [1]}, one obtains families of examples, see Example REF . One of our goals is to give true rank \(r\)  pairs and important contributions were made by Grinevich [2]}, Mokhov [3]}, [4]}, [5]}, [6]}, Mironov [7]}, Davletshina and Shamaev [8]}, Davletshina and Mironov [9]}, Mironov and Zheglov [10]}, [11]}, Oganesyan [12]}, [13]}, [14]}, Pogorelov and Zheglov [15]}. To check our results we constructed new true rank 2 pairs, by means of non self-adjoint operators of order 4 with genus 2 spectral curves, see Examples REF  and REF .
